The greater average pressure at the bottom of the stream is a natural consequence of the water falling. If you were to fountain the stream upwards, you would find the pressure near the top of the jet to be far lower than at the nozzle, not surprisingly. Up to a point this should be your experience, no matter what the pressure, but in practice, if the nozzle pressure is high enough the gravitational acceleration is too small to notice and interference from the air tends to break the jet up into a spray and slow its speed.

At a given rate of steady flow you are quite correct about the stream decreasing in diameter in compensation for the gravitational acceleration. The tap only supplies so many cc per second, and if the flow speed quadruples, then there is only enough water present at any instant to provide a stream of half the diameter. That is a matter of simple arithmetic.

There are however practical complications. You correctly mention air resistance as one example, but even without that there are several illusions, some of which you can explore with a digital camera in very bright light, or using a flash in the dark. You will find that a typical tap, when there is a steady pressure and it is open just wide enough, will give you an apparently smooth, tapering string of water from the nozzle downward for up to a foot or so. If you open it a bit wider you get an apparently solid, but clearly turbulent flow, splashy and noisy, and not tapering. The camera will show you that the smooth string really is smooth, but grows more variable in thickness the further it falls, until it develops regularly spaced constrictions that eventually split the stream into a sequence of separate drops, typically after a foot or so. The apparently turbulent flow on the other hand is illusory; it really is made up of a sequence of large blobs of water at regular intervals, so regular that you can freeze the flow with a suitably adjusted strobe light. It gives an eerie impression of writhing blobs suspended in midair.

The upper limit on the speed at which the stream would remain continuous depends on several factors but cannot exceed a few metres per second. Large drops of say, over 5 mm diameter, tend to break up at perhaps 10 m/s. The upper limit in a vacuum for a falling liquid would be somewhat over 11 km/s, that is to say escape velocity for Earth. But that too would involve several technical problems.

I submit that in a vacuum (within Newtonian velocities) that there is no limit.

But I am often wrong and nothing makes me happier than to be corrected.

And BTW one of the most "counter-intuitive" physics is that the pressure "drops" in the throat of a Venturi.

http://en.wikipedia.org/wiki/Venturi_effect

I have been playing with my kitchen tap. I should mention first of all that the tap does have an aerator which breaks the stream up into many small streams. With a low flow rate the stream does taper inward as it falls which I would expect as the speed of the stream increases as it falls. At high flow rates the reverse happens with the stream becoming wider as it falls. In this case I think wat is happening is that the sped of the water leaving the tap is higher than the terminal velocity and so the drag from the air is slowing the stream down.

I think that you will find that in a vacuum the water will boil producing an expanding cloud of steam that will eventually freeze into a cloud of ice crystals. I may be wrong but I am fairly sure that this is how freeze drying works.

Stewart, you are exacty right about the effect of vacuum on water of course. I should have mentioned that I was ignoring vapour pressure effects for clarity. For the sake of the demonstration I was thinking in terms of a high-vacuum oil!  :-)

Freeze drying can work as you describe when you are working with a liquid, in which case one usually starts by brute-force spraying the fluid into minute droplets, such as for producing milk powder or the like. However it might begin with ready-frozen water, or water too cool to explode into droplets, or too viscous to spray, quietly boiling itself  into  freezing until it must continue the process by sublimation. That can happen for example when freeze-drying fruit.